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Properties

Label 1-59-59.10-r1-0-0
Degree $1$
Conductor $59$
Sign $0.302 + 0.953i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.725 + 0.687i)2-s + (0.976 + 0.214i)3-s + (0.0541 + 0.998i)4-s + (−0.161 − 0.986i)5-s + (0.561 + 0.827i)6-s + (0.468 + 0.883i)7-s + (−0.647 + 0.762i)8-s + (0.907 + 0.419i)9-s + (0.561 − 0.827i)10-s + (0.994 + 0.108i)11-s + (−0.161 + 0.986i)12-s + (−0.907 + 0.419i)13-s + (−0.267 + 0.963i)14-s + (0.0541 − 0.998i)15-s + (−0.994 + 0.108i)16-s + (0.468 − 0.883i)17-s + ⋯
L(s)  = 1  + (0.725 + 0.687i)2-s + (0.976 + 0.214i)3-s + (0.0541 + 0.998i)4-s + (−0.161 − 0.986i)5-s + (0.561 + 0.827i)6-s + (0.468 + 0.883i)7-s + (−0.647 + 0.762i)8-s + (0.907 + 0.419i)9-s + (0.561 − 0.827i)10-s + (0.994 + 0.108i)11-s + (−0.161 + 0.986i)12-s + (−0.907 + 0.419i)13-s + (−0.267 + 0.963i)14-s + (0.0541 − 0.998i)15-s + (−0.994 + 0.108i)16-s + (0.468 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.302 + 0.953i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.302 + 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.491784367 + 1.824279621i\)
\(L(\frac12)\) \(\approx\) \(2.491784367 + 1.824279621i\)
\(L(1)\) \(\approx\) \(1.866652368 + 0.9098685522i\)
\(L(1)\) \(\approx\) \(1.866652368 + 0.9098685522i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.725 + 0.687i)T \)
3 \( 1 + (0.976 + 0.214i)T \)
5 \( 1 + (-0.161 - 0.986i)T \)
7 \( 1 + (0.468 + 0.883i)T \)
11 \( 1 + (0.994 + 0.108i)T \)
13 \( 1 + (-0.907 + 0.419i)T \)
17 \( 1 + (0.468 - 0.883i)T \)
19 \( 1 + (-0.856 - 0.515i)T \)
23 \( 1 + (0.370 - 0.928i)T \)
29 \( 1 + (-0.725 + 0.687i)T \)
31 \( 1 + (0.856 - 0.515i)T \)
37 \( 1 + (-0.647 - 0.762i)T \)
41 \( 1 + (-0.370 - 0.928i)T \)
43 \( 1 + (0.994 - 0.108i)T \)
47 \( 1 + (0.161 - 0.986i)T \)
53 \( 1 + (-0.561 - 0.827i)T \)
61 \( 1 + (0.725 + 0.687i)T \)
67 \( 1 + (-0.647 + 0.762i)T \)
71 \( 1 + (-0.161 + 0.986i)T \)
73 \( 1 + (-0.267 + 0.963i)T \)
79 \( 1 + (0.976 - 0.214i)T \)
83 \( 1 + (-0.796 + 0.605i)T \)
89 \( 1 + (0.725 - 0.687i)T \)
97 \( 1 + (-0.267 - 0.963i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.05326923811800145649235451808, −30.92623203567064744851365827310, −29.92493109197242463056653512416, −29.76895554973894482678945180122, −27.56342462772354489070994372164, −26.81261473538932240448959863105, −25.38442800469309364409328950572, −24.17038922113008151687289235494, −23.10433607350085811025398230530, −21.874463700898203171766482339127, −20.80913186344954815390812433607, −19.55055768061221666365458704394, −19.09906619538008720348381605123, −17.41705448296808735807595283262, −15.127083715145499381383924620071, −14.52598992916204698918125319551, −13.60524510810699415895104152529, −12.18386288297102932146589227048, −10.751690703697532045625626329470, −9.715176392977078085382622490515, −7.77132446483562033395164298409, −6.46913693155575063227031735781, −4.22394489429208307433994167439, −3.1956345512451963673833561902, −1.60693535295582980928122261665, 2.32680105920542059120146950219, 4.147764217249254539370128773090, 5.16769831829409609482848245929, 7.12028904535717930439022147625, 8.529528425013042958640315483785, 9.225325982233792702235976418831, 11.84378115804999157965923673625, 12.77494475957610797809233255717, 14.19906470866435276364076423183, 15.00040326116969592675489023668, 16.15639613595977496230172862084, 17.28133846736478968410322917501, 19.08820951812190086852002688846, 20.45413075949569233722986880155, 21.299084850162274333962953349291, 22.359058572813523714036853571224, 24.153380185219213958396168192194, 24.6993185597097385905963823771, 25.53405807224684594495916747730, 26.965854583430132595489318156778, 27.89640817041230508380202927950, 29.745193424486556185739839949242, 30.94483125914528319395930022573, 31.77717435513124302458784334667, 32.3430046399430264540868006790

Graph of the $Z$-function along the critical line